Date: Feb 11, 2013 2:58 AM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s
On 2/11/2013 1:53 AM, Virgil wrote:> In article <i6adnS7u2oO04oXMnZ2dnUVZ_uWdnZ2d@giganews.com>,> fom <fomJUNK@nyms.net> wrote:>>> On 2/10/2013 6:30 PM, Virgil wrote:>>> In article <hoqdnWmpiaOtvYXMnZ2dnUVZ_s2dnZ2d@giganews.com>,>>> fom <fomJUNK@nyms.net> wrote:>>>>>>> On 2/10/2013 4:16 PM, Virgil wrote:>>>>> In article>>>>> <3a8b891b-172f-415f-b4f6-34f988abae5d@e10g2000vbv.googlegroups.com>,>>>>> WM <mueckenh@rz.fh-augsburg.de> wrote:>>>>>>>>>>> On 10 Feb., 18:40, William Hughes <wpihug...@gmail.com> wrote:>>>>>>> On Feb 10, 10:51 am, WM <mueck...@rz.fh-augsburg.de> wrote:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 9 Feb., 17:36, William Hughes <wpihug...@gmail.com> wrote:>>>>>>>>>>>>>>>>>> the arguments are yours>>>>>>>>>>> and the statements are yours->>>>>>>>>>>>>>>>> Of course. But the wrong interpretation is yours.>>>>>>>>>>>>>>>> How does one interpret>>>>>>>>> we have shown m does not exist>>>>>>>>> (your statement)>>>>>>>>>>>>>>>> to mean that>>>>>>>>>>>>>>>> m might still exist>>>>>>>>>>>>>>>> ?>>>>>>>>>>>>>>> TND is invalid in the infinite.>>>>>>>>>>>>>>> Regards, WM>>>>>>>>>>>>>> In Wolkenmeukenheim, we can have>>>>>>> for a potentially infinite set>>>>>>>>>>>>>> we know that x does not exist>>>>>>> we don't know that x does not exist>>>>>>>>>>>>>> true at the same time.>>>>>>>>>>>> Is it so hard to conclude from facts without believing in matheology?>>>>>>>>>>>> The diagonal of the list>>>>>> 1>>>>>> 11>>>>>> 111>>>>>> ...>>>>>>>>>>>> is provably not in a particular line.>>>>>> But the diagonal is in the list, since it is defined in the list only.>>>>>> Nothing of the diagonal can be proven to surpass the lines and rows of>>>>>> the list.>>>>>>>>>> It is not that the diagonal "surpasses" any particular line, it is>>>>> merely that an appropriately defined "diagonal" is different from each>>>>> and every particular line, i.e., does not appear as any line among the>>>>> lines being listed.>>>>>>>> Yes. And the scare quotes are nice.>>>>>>>> The problem with singular terms means that>>>> "diagonal" is, in fact, a plurality of acts>>>> of definition.>>>>>> The Cantor antidiagonal rule, for an actually infinite list of actually>>> infinite binary sequences is a quite finite rule :>>>>>> If the two possible values are 'm' and 'w', then the nth term of the>>> diagonal is to be not equal to the nth term of the nth listed sequence,>>> meaning that>>> if the nth term of the nth listed sequence is "m">>> then the nth listed element of the diagonal is "w">>> and>>> if the nth term of the nth listed sequence is "w">>> then the nth listed element of the diagonal is "m".>>>>>> In this way, the constructed sequence differs from the nth listed>>> sequences at lest at its nth postion>>>>>>>>> Thanks, I do understand that.>>>> I was referring to WM's position. There cannot be one>> diagonal for him. Given n, WM must find a diagonal>> (note the indefinite article) such that length(dFIS)>n+1>> so that comparison with the n-th listed sequence can>> be made.>>>> While there may be other sources for the definition>> of "distinguishability", the one I have is in a book>> on automata. Distinguishability is characterized in>> terms of finitary "experiments of length k". Two>> "states" are k-distinguishable if there is an experiment>> of length k which differentiates them. Two states>> are distinguishable if they are k-distinguishable>> for any k.>> Shouldn't that be "k-distinguishable for some k"?

"some" is certainly better

while writing I must have had "any (particular)"in mind

>>>> Two "states" are k-equivalent if there is no m<=k for>> which the given states are differentiated by an experiment>> of length m.>>>> Two "states" are equivalent if for every k they are>> not k-distinguishable. So, equivalence is infinitary.>>>> This description coincides with your explanation>> as the Cantor diagonal is formed specifically to>> be k-distinguishable for every k.>>>> As for WM, definite articles imply representation>> with singular terms. He has a plural multiplicity>> of diagonals.> No one of which is the real one.>